archives@tulane.edu / In this thesis, we examine models of scale invariant behavior in univariate, multivariate, and high-dimensional settings from the viewpoint of wavelet-based statistical inference, and construct a new class of models called operator fractional Lévy motion. The first part of this work pertains to tempered fractional Brownian motion (tfBm), a model that displays transient scale invariant behavior. We use wavelets to construct the first estimation procedure for tfBm as well as a simple and computationally efficient hypothesis test and study their properties. In the second part of this thesis, we construct a new class of non-Gaussian second-order scale invariance models called operator fractional Lévy motion (ofLm) and study its probabilistic behavior. We then study asymptotic properties of wavelet eigenanalysis estimation applied to ofLm and examine its performance. In the last portion of this work, we study the mathematical framework of wavelet eigenanalysis in a multivariate setting with a view towards high-dimensional scale invariance modeling. We then proceed to conduct wavelet-based eigenanalysis in a high-dimensional setting, and conclude with some computational experiments. / 1 / Benjamin Boniece

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Thesis (Ph. D.)--University of Adelaide, Dept. of Applied Mathematics, 1987. / Includes bibliographical references (leaves 146-152).

Thesis (Ph. D.)--University of Florida, 1988. / Description based on print version record. Typescript. Vita. Includes bibliographical references.

Thesis--Wisconsin. / Vita. Includes bibliographical references (leaves 142-147).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1958. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaf [109]).

The general theory of stochastic optimal control is based on determining a control which minimizes an expected cost. However, the use of minimum expected cost as a design objective is arbitrary. A direct consequence of this choice is the need for extensive statistical information. If the required statistical data is not available or not accurate, the controller is suboptimum. The thesis begins with the investigation of the conventional method of solution and proposes an interpretation of the solution which introduces a different approach. This approach does not use the expected cost as design objective. The suggested new criterion is based on a trade-off between deterministic optimization and a cost penalty for estimation error. In order to have a basis of comparison with the conventional method, the proposed adaptive stochastic controller is compared with the standard stochastic optimal controller for a linear discrete system associated with linear measurements, additive noise and quadratic cost. The basic feature of the proposed method is the introduction of an adaptive filter gain which enters the proposed cost index algebraically and couples the controller with the estimator. Unlike the conventional Kalman-Bucy filter gain, the proposed gain is a scalar independent of the second and higher order moments of noise distributions. Simulation is carried out on second and fifth order linear systems with gaussian and non gaussian noises distributions. There is a moderate cost increase of 1% to 12%. The method is then extended to nonlinear systems. A general solution of the nonlinear problem is formulated and a complete investigation of the properties of the solution is given for different cases. Stability of the expected tracking error of the filter is guaranteed by introducing bounds on the filter gain. Problems arising from the use of suboptimum structures for the control are examined and discussed. It is shown that for a class of systems the proposed method has a particularly attractive form. As in the linear case, the required statistical information is limited to the expected values of the noises, and the expected value of the initial state of the system. Simulation executed on second order systems indicates a cost decrease of 1% to 20% when compared with the method using an extended Kalman-Bucy filter. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

A general method of analysis of a variety of stochastic processes in terms of probability density functions (PDFs) is developed and applied to several model as well as physically realistic systems. A model for diffusion in a bistable potential is the first system considered. The time dependence of the PDF for this system is described by a Fokker-Planck equation with non-linear coefficients. A numerical procedure is developed for finding the solution of this class of Fokker-Planck equations. The solution of the Fokker-Planck equation is obtained in terms of an eigenfunction expansion. The numerical procedure provides an efficient method of determining the eigenfunctions and eigenvalues of Fokker-Planck operators. The methods developed in the study of the model system are then applied to the trans-gauche isomerization of n-butane in CC1₄. This system is studied with the use of Kramers equation to describe the time evolution of the PDF. It is found that at room temperature the isomerization rate obeys a first order rate law. The rate constant for this system is sensitive to the collision frequency between the the n-butane and CC1₄ as has been previously suggested. It is also found that transition state theory underestimates the rate constant at all collision frequencies. However, the activation energy given by transition state theory is consistent with the activation energy obtained in this work. The problem of the escape of light constituents from planetary atmospheres is also considered. Here, the primary objective is to construct a collisional kinetic theory of planetary exospheres based on a rigorous solution of the Boltzmann equation. It is shown that this problem has many physical and mathematical similarities with the problems previously considered. The temperature and density profiles of light gases in the exosphere as well as their escape fluxes are calculated. In the present work, only a thermal escape mechanism was considered, although it is shown how non-thermal escape mechanisms may be included. In addition, these results are compared with various Monte-Carlo calculations of escape fluxes. / Science, Faculty of / Chemistry, Department of / Graduate